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Big mapping class groups - lecture 3

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Authors : Calegari, Danny (Author of the conference)
CIRM (Publisher )

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Abstract : Part I - Theory : In the "theory" part of this mini-course, we will present recent objects and phenomena related to the study of big mapping class groups. In particular, we will define two faithful actions of some big mapping class groups. The first is an action by isometries on a Gromov-hyperbolic graph. The second is an action by homeomorphisms on a circle in which the vertices of the graph naturally embed. We will describe some properties of the objects involved, and give some fruitful relations between the dynamics of the two actions. For example, we will see that loxodromic elements (for the first action) necessarily have rational rotation number (for the second action). Using these relations, we will explain how to construct non trivial quasimorphisms on subgroups of big mapping class groups. This includes joint work with Alden Walker.
Part II - Examples : In this part we will discuss a number of natural examples in which big mapping class groups and their subgroups arise. These include the inverse limit constructions of de Carvalho-Hall, the theory of finite depth (taut) foliations of 3-manifolds, the theory of “Artinization” of Thompson-like groups, two dimensional smooth dynamics, one dimensional complex dynamics (topology of the shift locus, Schottky spaces) and several other contexts. We will try to indicate how viewing these examples from the perspective of (big) mapping class groups is a worthwhile approach.

Keywords : mapping class groups; hyperbolic geometry; complex dynamics

MSC Codes :
37FXX - complex dynamical systems
57Mxx - Low-dimensional topology

Additional resources :
http://math.uchicago.edu/~dannyc/books/big_mcg/big_mcg.html

    Information on the Video

    Film maker : Recanzone, Luca
    Language : English
    Available date : 11/06/2019
    Conference Date : 22/05/2019
    Subseries : Research School
    arXiv category : Geometric Topology ; Dynamical Systems
    Mathematical Area(s) : Dynamical Systems & ODE ; Topology
    Format : MP4 (.mp4) - HD
    Video Time : 00:56:52
    Targeted Audience : Researchers
    Download : https://videos.cirm-math.fr/2019-05-22_Calegari_Part3.mp4

Information on the Event

Event Title : Dynamique au-delà de l'hyperbolicité uniforme / Dynamics Beyond Uniform Hyperbolicity
Event Organizers : Bonatti, Christian ; Buzzi, Jérôme ; Crovisier, Sylvain ; Gan, Shaobo ; Pacifico, Maria José
Dates : 13/05/2019 - 24/05/2019
Event Year : 2019
Event URL : https://conferences.cirm-math.fr/1947.html

Citation Data

DOI : 10.24350/CIRM.V.19526003
Cite this video as: Calegari, Danny (2019). Big mapping class groups - lecture 3. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19526003
URI : http://dx.doi.org/10.24350/CIRM.V.19526003

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